4 The two-slit experiment revisited

Under the conditions stipulated by Rule A, as we have seen, the probability of detection at the position D of a detector is the sum of two probabilities, pL = |AL|2 and pR = |AR|2. This is consistent with the view that an electron detected at D went through either L or R.

Setup of the two-slit experiment

Let us try to understand what happened, under the conditions stipulated by Rule B. Let us assume, to begin with, that

each electron goes through a particular slit (either L or R), and

the behavior of electrons that go through a given slit does not depend on whether the other slit is open or shut.

If the first assumption is true and both slits are open, the distribution of hits across the backdrop is given by

n(x) = nL(x) + nR(x),

where nL(x) and nR(x) are the respective distributions of hits from electrons that went through L and electrons that went through R. If the second assumption is true, then we can observe nL(x) by keeping the right slit shut, and we can observe nR(x) by keeping the left slit shut. What we observe if the right slit is shut is the dotted hump on the left side of Figure 1.4.2 (reproduced here), and what we observe when the left slit is shut is the dotted hump on the right side of this figure. If both assumptions are true, we thus expect to observe the sum of these two humps.

The probability of detection according to Rule A (the solid curve) is the sum of two probability distributions (the dotted curves), one for electrons that went through L and one for electrons that went through R.

But this is what we observe under the conditions stipulated by Rule A. What we observe under the conditions stipulated by Rule B is plotted in Figure 1.4.3. At least one of the two assumptions is therefore false.

The probability of detection according to Rule B.

Bohmian mechanics

According to an interpretational strategy proposed by David Bohm,[1] only the second assumption is false: all electrons follow well-defined paths, which wiggle in a peculiar manner and cluster at the backdrop so as to produce the observed distribution of hits.

Bohmian trajectories

What causes the wiggles? Bohmians explain this by positing the existence of a “pilot wave” that guides the electrons by exerting on them a force. If both slits are open, this passes through both slits; the secondary waves emanating from the slits interfere, and the result is that the electrons are guided along wiggly paths.

According to the Bohmians, the reason why electrons emerging from the same source or slit arrive in different places is that they start out in slightly different directions and/or with slightly different speeds. If we had precise knowledge of these initial values, we would be in a position to predict each electron’s future motion with classical precision. Since quantum mechanics decrees that such knowledge cannot be had, Bohmians must declare that in spite of the fact that well-defined electron paths and exact initial values exist, they are hidden from us. What they don’t say is why they are hidden from us, even though there is a perfectly simple answer to this question: they are hidden from us because they do not exist.

Bohmian mechanics is an extreme instantiation of the principle of evolution. It not only posits a wave function that evolves between measurements but also attributes to it the reality of a classical force that acts on classical particles, in blithe disregard of the fact that the pilot wave associated with a physical system with N degrees of freedom propagates in an N-dimensional configuration space, which can be identified with physical space only in the spacial case that N=3. (Another unpalatable feature of Bohmian mechanics is that on this theory energy and momentum and spin and every particle property other than position are contextual.[2])

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The meaning of “both”

Our interpretational strategy leads us to conclude that it is the first assumption which is wrong. Under the conditions stipulated by Rule B, each electron does in some sense pass through both slits. But in what sense? Saying that an electron went through both slits cannot be equivalent to saying that the electron went through L and that it went through R. To ascertain the truth of a conjunction we must individually ascertain the truths of its components, yet we never find (i) that an electron launched at G and detected at D has taken the left slit and (ii) that the same electron has taken the right slit.

Nor can saying that an electron went through both slits mean that a part of the electron went through L while another part went through R. In point of fact, the question of parts does not arise. Analogous experiments have been performed with C60 molecules using a grating with slits 50 nanometers wide and spaced 100 nanometers apart.[3] The sixty carbon nuclei of C60 are arranged like the corners of an old-fashioned soccer ball having a diameter of just 0.7 nanometers. We do not picture parts of such a molecule as getting separated by many times 100 nanometers and then reassemble into a ball less than a nanometer across.

A Buckminsterfullerene (C60)

Saying that an electron went through both slits can only mean that it went through L&R — the two elongated cutouts in the slit plate considered as an undifferentiated/undivided whole. Whenever Rule B applies, the distinction we make between L and R is a distinction that has no reality as far as the electron is concerned. The distinction between “the electron went through L” and “the electron went through R” is a distinction that “Nature does not make” — it corresponds to nothing in the actual world. The position at which the electron passed the slit plate is the entire undifferentiated region L&R. It is not any part or segment of L&R, let alone a point somewhere in L&R.